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(cos(x)^2-cos(2*x)^2)/x^2
Trazado el gráfico de la función/ (cos(x)^2-cos(2*x)^2)/x^2

Gráfico de la función y = (cos(x)^2-cos(2*x)^2)/x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2         2     
       cos (x) - cos (2*x)
f(x) = -------------------
                 2        
                x
f(x)=cos⁡2(x)−cos⁡2(2x)x2f{\left(x \right)} = \frac{\cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)}}{x^{2}} 
f = (cos(x)^2 - cos(2*x)^2)/x^2
Gráfico de la función
02468-8-6-4-2-10105-5
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
cos⁡2(x)−cos⁡2(2x)x2=0\frac{\cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−5π3x_{1} = - \frac{5 \pi}{3}
x2=−4π3x_{2} = - \frac{4 \pi}{3}
x3=−πx_{3} = - \pi
x4=−2π3x_{4} = - \frac{2 \pi}{3}
x5=−π3x_{5} = - \frac{\pi}{3}
x6=π3x_{6} = \frac{\pi}{3}
x7=2π3x_{7} = \frac{2 \pi}{3}
x8=πx_{8} = \pi
x9=4π3x_{9} = \frac{4 \pi}{3}
x10=5π3x_{10} = \frac{5 \pi}{3}
x11=2πx_{11} = 2 \pi
Solución numérica
x1=−39.7935069454707x_{1} = -39.7935069454707
x2=56.5486676669743x_{2} = 56.5486676669743
x3=−26.1799387799149x_{3} = -26.1799387799149
x4=48.1710873550435x_{4} = 48.1710873550435
x5=−21.9911485864806x_{5} = -21.9911485864806
x6=−75.3982234033138x_{6} = -75.3982234033138
x7=78.5398162405851x_{7} = 78.5398162405851
x8=−53.4070748816993x_{8} = -53.4070748816993
x9=−41.8879020478639x_{9} = -41.8879020478639
x10=90.0589894029074x_{10} = 90.0589894029074
x11=4.18879020478639x_{11} = 4.18879020478639
x12=12.5663704006205x_{12} = 12.5663704006205
x13=−94.2477794994859x_{13} = -94.2477794994859
x14=−9.42477795811289x_{14} = -9.42477795811289
x15=−9.42477801944606x_{15} = -9.42477801944606
x16=43.9822971691727x_{16} = 43.9822971691727
x17=−28.2743337735251x_{17} = -28.2743337735251
x18=26.1799387799149x_{18} = 26.1799387799149
x19=52.3598775598299x_{19} = 52.3598775598299
x20=19.8967534727354x_{20} = 19.8967534727354
x21=8.37758040957278x_{21} = 8.37758040957278
x22=72.2566310277254x_{22} = 72.2566310277254
x23=50.2654824463829x_{23} = 50.2654824463829
x24=−59.6902604565858x_{24} = -59.6902604565858
x25=−2.0943951023932x_{25} = -2.0943951023932
x26=−87.9645942863265x_{26} = -87.9645942863265
x27=24.0855436775217x_{27} = 24.0855436775217
x28=−37.699111876364x_{28} = -37.699111876364
x29=−90.0589894029074x_{29} = -90.0589894029074
x30=68.0678408277789x_{30} = 68.0678408277789
x31=6.28318528411306x_{31} = 6.28318528411306
x32=−50.2654823495157x_{32} = -50.2654823495157
x33=−6.28318518064309x_{33} = -6.28318518064309
x34=−72.2566309245122x_{34} = -72.2566309245122
x35=−24.0855436775217x_{35} = -24.0855436775217
x36=−15.7079632958702x_{36} = -15.7079632958702
x37=−68.0678408277789x_{37} = -68.0678408277789
x38=−97.3893723008071x_{38} = -97.3893723008071
x39=28.274333865278x_{39} = 28.274333865278
x40=126.710903694788x_{40} = 126.710903694788
x41=−70.162235930172x_{41} = -70.162235930172
x42=98.4365698124802x_{42} = 98.4365698124802
x43=17.8023583703422x_{43} = 17.8023583703422
x44=−85.870199198121x_{44} = -85.870199198121
x45=37.6991119346587x_{45} = 37.6991119346587
x46=−79.5870138909414x_{46} = -79.5870138909414
x47=−53.407074872779x_{47} = -53.407074872779
x48=−46.0766922526503x_{48} = -46.0766922526503
x49=−13.6135681655558x_{49} = -13.6135681655558
x50=−4.18879020478639x_{50} = -4.18879020478639
x51=60.7374579694027x_{51} = 60.7374579694027
x52=−32.4631240870945x_{52} = -32.4631240870945
x53=94.2477796093531x_{53} = 94.2477796093531
x54=41.8879020478639x_{54} = 41.8879020478639
x55=34.557518437728x_{55} = 34.557518437728
x56=30.3687289847013x_{56} = 30.3687289847013
x57=−9.42477970152278x_{57} = -9.42477970152278
x58=39.7935069454707x_{58} = 39.7935069454707
x59=−83.7758040957278x_{59} = -83.7758040957278
x60=−65.9734457654187x_{60} = -65.9734457654187
x61=−81.6814090366049x_{61} = -81.6814090366049
x62=100.530964814456x_{62} = 100.530964814456
x63=−17.8023583703422x_{63} = -17.8023583703422
x64=213.628236456552x_{64} = 213.628236456552
x65=−43.9822971747433x_{65} = -43.9822971747433
x66=−48.1710873550435x_{66} = -48.1710873550435
x67=46.0766922526503x_{67} = 46.0766922526503
x68=81.6814090780798x_{68} = 81.6814090780798
x69=21.9911485851203x_{69} = 21.9911485851203
x70=−287.979326579064x_{70} = -287.979326579064
x71=−63.8790506229925x_{71} = -63.8790506229925
x72=−61.7846555205993x_{72} = -61.7846555205993
x73=63.8790506229925x_{73} = 63.8790506229925
x74=−92.1533845053006x_{74} = -92.1533845053006
x75=2.0943951023932x_{75} = 2.0943951023932
x76=74.3510261349584x_{76} = 74.3510261349584
x77=59.6902605072247x_{77} = 59.6902605072247
x78=83.7758040957278x_{78} = 83.7758040957278
x79=−97.3893723371052x_{79} = -97.3893723371052
x80=12.5663705120713x_{80} = 12.5663705120713
x81=−87.9645943594843x_{81} = -87.9645943594843
x82=96.342174710087x_{82} = 96.342174710087
x83=−31.4159266036064x_{83} = -31.4159266036064
x84=69.1150381715544x_{84} = 69.1150381715544
x85=54.4542726622231x_{85} = 54.4542726622231
x86=−57.5958653158129x_{86} = -57.5958653158129
x87=−19.8967534727354x_{87} = -19.8967534727354
x88=85.870199198121x_{88} = 85.870199198121
x89=−53.4070751728923x_{89} = -53.4070751728923
x90=−75.3982237384571x_{90} = -75.3982237384571
x91=15.7079633572691x_{91} = 15.7079633572691
x92=87.964594334847x_{92} = 87.964594334847
x93=−69.1150384491256x_{93} = -69.1150384491256
x94=76.4454212373516x_{94} = 76.4454212373516
x95=65.9734457523646x_{95} = 65.9734457523646
x96=92.1533845053006x_{96} = 92.1533845053006
x97=70.162235930172x_{97} = 70.162235930172
x98=−35.6047167406843x_{98} = -35.6047167406843
x99=34.5575190929181x_{99} = 34.5575190929181
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (cos(x)^2 - cos(2*x)^2)/x^2.
−cos⁡2(0⋅2)+cos⁡2(0)02\frac{- \cos^{2}{\left(0 \cdot 2 \right)} + \cos^{2}{\left(0 \right)}}{0^{2}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡2(x)−cos⁡2(2x)x2)=0\lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡2(x)−cos⁡2(2x)x2)=0\lim_{x \to \infty}\left(\frac{\cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (cos(x)^2 - cos(2*x)^2)/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡2(x)−cos⁡2(2x)xx2)=0\lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)}}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡2(x)−cos⁡2(2x)xx2)=0\lim_{x \to \infty}\left(\frac{\cos^{2}{\left(x \right)} - \cos^{2}{\left(2 x \right)}}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Gráfico
Gráfico de la función y = (cos(x)^2-cos(2*x)^2)/x^2
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